The following work have been accepted for publication in "Nonlinear Analysis: Real World Applications" and is now available online:

Well-posedness on a large Besov-weak-Morrey space and novel numerical insights for the inviscid SQG equation

https://doi.org/10.1016/j.nonrwa.2026.104645

E. Abreu, J. Pérez, N. Quispe-Cuba, J. C. Valencia-Guevara

Abstract:

In this paper, we develop a mathematical framework for generating novel estimates of Bernstein type, commutator inequalities, and Logarithmic type inequality in the modified Besov-weak-Morrey spaces to allow us to get the local in time well-posedness for the inviscid Surface Quasi-Geostrophic (SQG) equation via the method of successive approximations. The purpose of this paper is two-fold. First, we prove local-in-time well-posedness theory for the inviscid SQG equation in a large class of Besov-weak-Morrey type, improving recent results, covering now sharp results for critical and sub-critical regularity cases. In addition, we also present the criterion of blow-up for the SQG equation in this new framework. We also obtain a key estimate, which is new in our framework, as far as we know, which allows us to recover several well-known key results for the classic Besov spaces, such as Sobolev-like immersion and Log-Sobolev-type inequalities. Second, we show and present a qualitative discussion, based on numerical insights for the dynamics of the solution for the inviscid SQG. The novelty is a possible self-organization, as time evolves, of non-singular solutions into pairs of positive-negative vorticities, which continue evolving following a pair-trajectory. The novel approach has been developed in a non-periodic setting, leading solutions to travel beyond possible boundary periodic conditions, which allow us to capture the formation of strong fronts, also called frontogenesis, preserving L1 and L2 norms, thereby adding new improvements on similarities with the 3D incompressible Euler equations, as observed by Constantin, Majda and Tabak, but now on unbounded domains.

Keywords:

Quasi-Geostrophic equations; Nonlocal fluxes; Riesz transform; Besov-type spaces; Volume-preserving map; Commutator estimates; Morrey-type spaces; Computer-assisted proofs; Numerical insights.

2010 MSC:

35Q31, 35A07, 76B03, 42B35, 46E30, 46E35.